Subgroups — C₃×#10602581 #14307378

Order 1920.

552 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#65225	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic
4	C₃×#64749	C₂	1	4	yes	maximal
5	#10602581	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
9	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂²×C₃×#168	C₂²	1	2	yes	—
14	C₃×#3963	—	2	1	no	—
15	C₃×#3963	—	2	1	no	—
16	C₃×#3935	—	2	2	no	—
17	C₃×#1993	—	5	1	no	maximal, Hall
18	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	#65225	C₂×C₃	1	1	yes	characteristic
20	C₂×#3935	C₂×C₃	1	1	yes	characteristic
21	#64749	C₂×C₃	1	4	yes	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	4	4	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₅	—	2	2	no	—
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₃×C₄⋊Dic₅	—	2	4	no	—
35	C₂×C₃×#168	—	2	2	no	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	Q₈	1	4	yes	complemented
38	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
39	C₂×C₃×#97	—	5	1	no	—
40	C₃×#382	—	5	1	no	—
41	C₃×#340	—	5	4	no	—
42	C₂²×C₅×Q₈	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
45	C₂×C₅×C₄⋊C₄	—	2	1	no	—
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
49	C₂²×#168	C₂²×C₃	1	2	yes	—
50	#3963	—	2	1	no	—
51	#3963	—	2	1	no	—
52	#3935	—	2	2	no	—
53	#1993	—	5	1	no	Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₃×C₅×Q₈	—	4	8	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
65	C₂×C₃×C₄×C₅	#41	1	2	yes	—
66	C₂×C₃×Dic₅	—	4	2	no	—
67	C₂×C₃×Dic₅	—	4	2	no	—
68	C₂×C₃×Dic₅	—	4	2	no	—
69	C₃×#168	—	2	4	no	—
70	C₂²×C₃×Q₈	—	2	1	no	—
71	C₂²×C₃×C₈	—	5	2	no	—
72	C₃×C₄⋊Q₈	—	2	2	no	—
73	C₃×C₄⋊Q₈	Dic₅	1	4	yes	complemented
74	C₂×C₃×C₄⋊C₄	—	2	1	no	—
75	C₂×C₃×C₄⋊C₄	—	5	2	no	—
76	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
77	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
78	C₃×#97	—	10	2	no	—
79	C₃×#95	—	10	1	no	—
80	C₃×#95	—	10	1	no	—
81	C₂×C₅×Q₈	—	2	4	no	—
82	C₂×C₅×Q₈	—	4	4	no	—
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₂²×Dic₅	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	4	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
91	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
92	C₄²×C₅	C₃×D₄	1	2	yes	—
93	C₄⋊Dic₅	—	2	4	no	—
94	C₂×#168	—	2	2	no	—
95	C₂×#168	—	2	2	no	—
96	C₂×#168	C₃×Q₈	1	4	yes	complemented
97	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic
98	C₂×#97	—	5	1	no	—
99	#382	—	5	1	no	—
100	#340	—	5	4	no	—
101	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	#124	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
104	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
105	C₂²×C₃×C₅	#137	1	2	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	2	no	—
108	C₃×C₄×C₅	—	4	4	no	—
109	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
110	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
111	C₃×Dic₅	—	8	2	no	—
112	C₂×C₃×Q₈	—	2	4	no	—
113	C₂×C₃×Q₈	—	4	4	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	10	2	no	—
116	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
119	C₂×C₃×C₈	—	5	4	no	—
120	C₂×C₃×C₈	—	10	2	no	—
121	C₂×C₃×C₈	—	10	2	no	—
122	C₃×C₄⋊C₄	—	2	4	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	—	10	4	no	—
125	C₃×C₄⋊C₄	C₂×Dic₅	1	4	yes	—
126	C₃×C₄²	C₂×Dic₅	1	2	yes	—
127	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
128	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
129	C₅×Q₈	—	4	8	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
139	C₂×C₄×C₅	C₃×#41	1	2	yes	—
140	C₂×Dic₅	—	4	2	no	—
141	C₂×Dic₅	—	4	2	no	—
142	C₂×Dic₅	—	4	2	no	—
143	#168	—	2	4	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂²×C₈	—	5	2	no	—
146	C₄⋊Q₈	—	2	2	no	—
147	C₄⋊Q₈	C₃×Dic₅	1	4	yes	complemented
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	5	2	no	—
150	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
151	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
152	#97	—	10	2	no	—
153	#95	—	10	1	no	—
154	#95	—	10	1	no	—
155	C₂×C₃×C₅	#391	1	1	yes	characteristic
156	C₂×C₃×C₅	#418	1	1	yes	characteristic
157	C₂×C₃×C₅	#419	1	1	yes	characteristic
158	C₂×C₃×C₅	#489	1	2	yes	—
159	C₂×C₃×C₅	#501	1	2	yes	—
160	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
161	C₃×Q₈	—	4	8	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	4	4	no	—
166	C₂×C₃×C₄	—	20	2	no	—
167	C₂×C₃×C₄	—	20	2	no	—
168	C₂×C₃×C₄	—	20	2	no	—
169	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
170	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
171	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
172	C₂×C₃×C₄	#713	1	2	yes	—
173	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
175	C₃×C₈	—	10	4	no	—
176	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₅	C₃×#124	1	1	yes	characteristic
178	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
179	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
180	C₂²×C₅	C₃×#137	1	2	yes	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	4	4	no	—
184	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
185	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
186	Dic₅	—	8	2	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	10	2	no	—
191	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
192	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
193	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
194	C₂×C₈	—	5	4	no	—
195	C₂×C₈	—	10	2	no	—
196	C₂×C₈	—	10	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	10	4	no	—
200	C₄⋊C₄	C₂×C₃×Dic₅	1	4	yes	—
201	C₄²	C₂×C₃×Dic₅	1	2	yes	—
202	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
203	C₃×C₅	#1993	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	#4078	1	2	yes	—
205	C₂²×C₃	#4088	1	2	yes	—
206	C₂²×C₃	#4090	1	1	yes	characteristic
207	C₂²×C₃	#4093	1	1	yes	characteristic
208	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	4	4	no	—
212	C₃×C₄	—	40	2	no	—
213	C₃×C₄	Q₈×Dic₅	1	2	yes	—
214	C₃×C₄	#4092	1	2	yes	—
215	C₂×C₅	C₃×#391	1	1	yes	characteristic
216	C₂×C₅	C₃×#418	1	1	yes	characteristic
217	C₂×C₅	C₃×#419	1	1	yes	characteristic
218	C₂×C₅	C₃×#489	1	2	yes	—
219	C₂×C₅	C₃×#501	1	2	yes	—
220	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
221	Q₈	—	4	8	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	20	2	no	—
227	C₂×C₄	—	20	2	no	—
228	C₂×C₄	—	20	2	no	—
229	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
230	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
231	C₂×C₄	C₃×Q₈×D₅	1	2	yes	—
232	C₂×C₄	C₃×#713	1	2	yes	—
233	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
234	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
235	C₈	—	10	4	no	—
236	C₂×C₃	#65369	1	2	yes	—
237	C₂×C₃	#65378	1	2	yes	—
238	C₂×C₃	#65504	1	1	yes	characteristic
239	C₂×C₃	#65514	1	1	yes	characteristic
240	C₂×C₃	#65519	1	1	yes	characteristic
241	C₅	C₃×#1993	1	1	yes	characteristic, complemented, Sylow
242	C₂²	C₃×#4078	1	2	yes	—
243	C₂²	C₃×#4088	1	2	yes	—
244	C₂²	C₃×#4090	1	1	yes	characteristic
245	C₂²	C₃×#4093	1	1	yes	characteristic
246	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	4	no	—
250	C₄	—	40	2	no	—
251	C₄	C₃×Q₈×Dic₅	1	2	yes	—
252	C₄	C₃×#4092	1	2	yes	—
253	C₃	#10602581	1	1	yes	characteristic, direct product factor, Sylow
254	C₂	C₃×#65369	1	2	yes	—
255	C₂	C₃×#65378	1	2	yes	—
256	C₂	C₃×#65504	1	1	yes	characteristic
257	C₂	C₃×#65514	1	1	yes	characteristic
258	C₂	C₃×#65519	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#1993
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
5. #10602581	C₃	direct (N × H)
8. C₃×C₅×C₄⋊Q₈	C₄	semidirect
37. C₂×C₃×#168	Q₈	semidirect
44. C₅×C₄⋊Q₈	C₃×C₄	semidirect
73. C₃×C₄⋊Q₈	Dic₅	semidirect
96. C₂×#168	C₃×Q₈	semidirect
147. C₄⋊Q₈	C₃×Dic₅	semidirect
203. C₃×C₅	#1993	semidirect
241. C₅	C₃×#1993	semidirect
253. C₃	#10602581	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
C₃×#65225	2
C₂×C₃×#3935	2
C₃×#64749	2
#10602581	3
C₃×#1993	5

Subgroups — C3×#10602581 #14307378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602581 #14307378